Question

suppose that two cards are randomly selected from a standard 52 - card deck.
click the icon to view a deck of cards
(a) what is the probability that the first card is a spade and the second card is a spade if the sampling is done without replacement?
(b) what is the probability that the first card is a spade and the second card is a spade if the sampling is done with replacement?
(a) if the sampling is done without replacement, the probability that the first card is a spade and the second card is a spade is (round to three decimal places as needed.)

Explanation:

Step1: Calculate probability of first - spade

In a standard 52 - card deck, there are 13 spades. The probability that the first card is a spade without replacement is $\frac{13}{52}=\frac{1}{4}$.

Step2: Calculate probability of second - spade

After drawing one spade, there are 12 spades left and 51 cards left. The probability that the second card is a spade given the first was a spade is $\frac{12}{51}$.

Step3: Calculate joint probability

By the multiplication rule for dependent events, the probability that the first card is a spade and the second card is a spade without replacement is $\frac{13}{52}\times\frac{12}{51}=\frac{13\times12}{52\times51}=\frac{156}{2652}\approx0.059$.

Step4: Calculate probability with replacement

If sampling is done with replacement, the probability that the first card is a spade is $\frac{13}{52}=\frac{1}{4}$, and the probability that the second card is a spade is also $\frac{13}{52}=\frac{1}{4}$ since the deck is restored to its original state. The joint probability is $\frac{13}{52}\times\frac{13}{52}=\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}= 0.0625$.

Answer:

(a) 0.059
(b) 0.063 (rounded to three decimal places)